Here is an integral that appears in the table of integrals by Gradshtein and Ryzhik, it was also studied by Ramanujan (not sure his original solution was found - it seems it doesn't appear in
any of the notebooks).

Now, by complex analysis, one can brifely finish it, I'm not interested in such a solution. But thinking of Ramanujan I'm sure he had
a solution using methods of real analysis (and to avoid
possible misunderstandings, I mean not even a touch on complex numbers - to be clear).

Do you know such a solution? Post it only if you want to, I'm only curious if such solutions are known, maybe some simple such solutions?

$\begingroup$Have you tried using the Fourier series expansion of $\sin(ax)$ ?$\endgroup$
– TheOscillatorMar 26 '16 at 20:37

$\begingroup$If it appears in G&R, it'd help to reference where in the text it shows up. Additionally, for what $a,b$ does this identity hold?$\endgroup$
– SemiclassicalMar 27 '16 at 20:31

$\begingroup$The properties of Beta and Gamma functions are closely related to complex analysis, and can't be easily obtained another way. So can we use those without proof?$\endgroup$
– Yuriy SAug 18 '16 at 8:02